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[,1] [,2] [,3] [,4]
[1,] "Coari x Yangambí" "Unipalma" "Amazon" "Coari x Lame"
[2,] "Coari x Yangambí" "Coari x Lame" "Unipalma" "Amazon"
[3,] "Coari x Yangambí" "Amazon" "Coari x Lame" "Unipalma"
[4,] "Coari x Lame" "Amazon" "Unipalma" "Coari x Yangambí"
[5,] "Unipalma" "Amazon" "Coari x Yangambí" "Coari x Lame"
Call:
lm(formula = rdt ~ trt + F, data = data)
Residuals:
Min 1Q Median 3Q Max
-5.0708 -1.0588 -0.0568 0.9452 3.0932
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 46.761 1.463 31.965 6.31e-16 ***
trtB 4.584 1.542 2.973 0.00898 **
trtC 8.594 1.542 5.573 4.20e-05 ***
trtD 12.034 1.542 7.804 7.64e-07 ***
trtE 11.744 1.542 7.616 1.04e-06 ***
FF2 2.152 1.542 1.396 0.18191
FF3 -1.864 1.542 -1.209 0.24430
FF4 -1.576 1.542 -1.022 0.32197
FF5 3.154 1.542 2.045 0.05762 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.438 on 16 degrees of freedom
Multiple R-squared: 0.8673, Adjusted R-squared: 0.8009
F-statistic: 13.07 on 8 and 16 DF, p-value: 1.091e-05
Conclusión: Se concluye que los coeficientes \(\beta_0\), \(\beta_1\), \(\beta_2\), \(\beta_3\), \(\beta_4\) correspondientes a los tratamientos B, C, D y E, fueron estadísticamente distintos de 0. El ajuste del modelo descrito en el \(R^2\) fue de 0.8673, lo que quiere decir que el 86.73 % de la varianza del rendimiento está explicado por las variables Tratamiento y Bloque.
Durbin-Watson test
data: modelo.dbca
DW = 1.0391, p-value = 0.07273
alternative hypothesis: true autocorrelation is not 0
Conclusión. A un nivel de significancia de 0.1, se concluye que existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, los residuos del rendimiento no son completamente aleatorios e independientes.
Normalidad de residuos
\(H_0: \text{La distribución de los residuos del rendimiento es similar a la función normal}\)
\(H_1: \text{La distribución de los residuos del rendimiento es similar a la función normal}\)
shapiro.test(rstudent(modelo.dbca))
Shapiro-Wilk normality test
data: rstudent(modelo.dbca)
W = 0.93794, p-value = 0.1327
ad.test(rstudent(modelo.dbca))
Anderson-Darling normality test
data: rstudent(modelo.dbca)
A = 0.40488, p-value = 0.328
lillie.test(rstudent(modelo.dbca))
Lilliefors (Kolmogorov-Smirnov) normality test
data: rstudent(modelo.dbca)
D = 0.14105, p-value = 0.2245
Exact one-sample Kolmogorov-Smirnov test
data: rstudent(modelo.dbca)
D = 0.11248, p-value = 0.8752
alternative hypothesis: two-sided
cvm.test(rstudent(modelo.dbca))
Cramer-von Mises normality test
data: rstudent(modelo.dbca)
W = 0.05227, p-value = 0.4635
pearson.test(rstudent(modelo.dbca))
Pearson chi-square normality test
data: rstudent(modelo.dbca)
P = 3.48, p-value = 0.6264
sf.test(rstudent(modelo.dbca))
Shapiro-Francia normality test
data: rstudent(modelo.dbca)
W = 0.93114, p-value = 0.08602
Conclusión. A un nivel de significancia de 0.1, se concluye que no existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, la distribución de los residuos del rendimiento es similar a la función normal o gaussiana.
Homocedasticidad
\(H_0\): La varianza del rendimiento es constante con respecto a los valores ajustados del rendimiento.
\(H_1\): La varianza del rendimiento no es constante con respecto a los valores ajustados del rendimiento.
ncvTest(modelo.dbca)
Non-constant Variance Score Test
Variance formula: ~ fitted.values
Chisquare = 2.80928, Df = 1, p = 0.09372
bptest(modelo.dbca)
studentized Breusch-Pagan test
data: modelo.dbca
BP = 11.967, df = 8, p-value = 0.1527
bptest(modelo.dbca, studentize = F)
Breusch-Pagan test
data: modelo.dbca
BP = 12.691, df = 8, p-value = 0.1229
olsrr::ols_test_breusch_pagan(modelo.dbca)
Breusch Pagan Test for Heteroskedasticity
-----------------------------------------
Ho: the variance is constant
Ha: the variance is not constant
Data
-------------------------------
Response : rdt
Variables: fitted values of rdt
Test Summary
-----------------------------
DF = 1
Chi2 = 2.80928
Prob > Chi2 = 0.09372041
Conclusión. A un nivel de significancia de 0.1, se concluye que no existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, la varianza del rendimiento es constante con respecto a los valores ajustados del rendimiento.
Recomendación. Debido a que se cumple con el supuesto de homocedasticidad, para evaluar los efectos de los tratamientos con respecto al rendimiento, se debe proceder a realizar el análisis de varianza.
Tukey's test of nonadditivity
rdt
P : -241.6241
Q : 8290.191
Analysis of Variance Table
Response: residual
Df Sum Sq Mean Sq F value Pr(>F)
Nonadditivity 1 7.042 7.0423 1.1994 0.2907
Residuals 15 88.069 5.8713
$P
Nonadditivity
-241.6241
$Q
Nonadditivity
8290.191
$ANOVA
Analysis of Variance Table
Response: residual
Df Sum Sq Mean Sq F value Pr(>F)
Nonadditivity 1 7.042 7.0423 1.1994 0.2907
Residuals 15 88.069 5.8713
Estadísticas globales
modelo.dbca %>%gvlma()
Call:
lm(formula = rdt ~ trt + F, data = data)
Coefficients:
(Intercept) trtB trtC trtD trtE FF2
46.761 4.584 8.594 12.034 11.744 2.152
FF3 FF4 FF5
-1.864 -1.576 3.154
ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
Level of Significance = 0.05
Call:
gvlma(x = .)
Value p-value Decision
Global Stat 2.94984 0.5663 Assumptions acceptable.
Skewness 0.54384 0.4608 Assumptions acceptable.
Kurtosis 0.01525 0.9017 Assumptions acceptable.
Link Function 1.85107 0.1737 Assumptions acceptable.
Heteroscedasticity 0.53969 0.4626 Assumptions acceptable.
Conclusión. A un nivel de significancia de 0.05, se concluye que existe suficiente evidencia estadística para rechazar la hipótesis nula, por lo tanto, al menos un tratamiento tiene un efecto sobre el rendimiento estadísticamente diferente del resto de tratamientos.
agricolae::cv.model(modelo.dbca)
[1] 4.471568
Comparaciones de medias
A vs B:
\(H_0: \mu_{A} - \mu_{B} = 0\)
\(H_1: \mu_{A} - \mu_{B} \neq 0\)
A vs C:
\(H_0: \mu_{A} - \mu_{C} = 0\)
\(H_1: \mu_{A} - \mu_{C} \neq 0\)
A vs D:
\(H_0: \mu_{A} - \mu_{D} = 0\)
\(H_1: \mu_{A} - \mu_{D} \neq 0\)
A vs E:
\(H_0: \mu_{A} - \mu_{E} = 0\)
\(H_1: \mu_{A} - \mu_{E} \neq 0\)
B vs C:
\(H_0: \mu_{B} - \mu_{C} = 0\)
\(H_1: \mu_{B} - \mu_{C} \neq 0\)
B vs D:
\(H_0: \mu_{B} - \mu_{D} = 0\)
\(H_1: \mu_{B} - \mu_{D} \neq 0\)
B vs E:
\(H_0: \mu_{B} - \mu_{E} = 0\)
\(H_1: \mu_{B} - \mu_{E} \neq 0\)
C vs D:
\(H_0: \mu_{C} - \mu_{D} = 0\)
\(H_1: \mu_{C} - \mu_{D} \neq 0\)
C vs E:
\(H_0: \mu_{C} - \mu_{E} = 0\)
\(H_1: \mu_{C} - \mu_{E} \neq 0\)
D vs E:
\(H_0: \mu_{D} - \mu_{E} = 0\)
\(H_1: \mu_{D} - \mu_{E} \neq 0\)
Prueba de HSD (Honestamente significativa -> Tukey)
agricolae::HSD.test(modelo.dbca, trt ="trt", alpha =0.05,group =TRUE, main =NULL, console =TRUE)
Study: modelo.dbca ~ "trt"
HSD Test for rdt
Mean Square Error: 5.944484
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 16
Critical Value of Studentized Range: 4.332688
Minimun Significant Difference: 4.724213
Treatments with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 ab
B 51.718 bc
A 47.134 c
agricolae::HSD.test(modelo.dbca, trt ="trt", alpha =0.05,group =FALSE, main =NULL, console =TRUE)
Study: modelo.dbca ~ "trt"
HSD Test for rdt
Mean Square Error: 5.944484
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 16
Critical Value of Studentized Range: 4.332688
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0594 . -9.308213 0.1402129
A - C -8.594 0.0004 *** -13.318213 -3.8697871
A - D -12.034 0.0000 *** -16.758213 -7.3097871
A - E -11.744 0.0000 *** -16.468213 -7.0197871
B - C -4.010 0.1173 -8.734213 0.7142129
B - D -7.450 0.0015 ** -12.174213 -2.7257871
B - E -7.160 0.0022 ** -11.884213 -2.4357871
C - D -3.440 0.2182 -8.164213 1.2842129
C - E -3.150 0.2909 -7.874213 1.5742129
D - E 0.290 0.9997 -4.434213 5.0142129
Nota: Todos aquellos tratamientos que compartan por lo menos una letra serán estadísticamente similares.
Recomendación: Se debe usar la prueba de Tukey cuando como máximo se tenga 6 tratamientos.
Prueba de Duncan
agricolae::duncan.test(modelo.dbca, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dbca ~ "trt"
Duncan's new multiple range test
for rdt
Mean Square Error: 5.944484
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 16
Critical Range
2 3 4 5
3.268914 3.427893 3.527272 3.595419
Means with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 ab
C 55.728 b
B 51.718 c
A 47.134 d
agricolae::duncan.test(modelo.dbca, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dbca ~ "trt"
Duncan's new multiple range test
for rdt
Mean Square Error: 5.944484
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0090 ** -7.852914 -1.31508590
A - C -8.594 0.0001 *** -12.021893 -5.16610657
A - D -12.034 0.0000 *** -15.629419 -8.43858057
A - E -11.744 0.0000 *** -15.271272 -8.21672833
B - C -4.010 0.0193 * -7.278914 -0.74108590
B - D -7.450 0.0003 *** -10.977272 -3.92272833
B - E -7.160 0.0004 *** -10.587893 -3.73210657
C - D -3.440 0.0493 * -6.867893 -0.01210657
C - E -3.150 0.0579 . -6.418914 0.11891410
D - E 0.290 0.8532 -2.978914 3.55891410
Nota: La prueba de Duncan usa múltiples valores críticos.
Recomendación: No usar la prueba de Duncan cuando se presentan resultados con múltiples tratamientos que obtengan varias significancias.
Prueba de Student - Newman -Keuls
agricolae::SNK.test(modelo.dbca, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dbca ~ "trt"
Student Newman Keuls Test
for rdt
Mean Square Error: 5.944484
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Alpha: 0.05 ; DF Error: 16
Critical Range
2 3 4 5
3.268914 3.978895 4.411720 4.724213
Means with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 a
C 55.728 a
B 51.718 b
A 47.134 c
agricolae::SNK.test(modelo.dbca, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dbca ~ "trt"
Student Newman Keuls Test
for rdt
Mean Square Error: 5.944484
trt, means
rdt std r Min Max
A 47.134 2.840129 5 42.26 49.43
B 51.718 5.324328 5 44.41 57.31
C 55.728 2.229747 5 53.72 59.45
D 59.168 2.065798 5 55.87 60.89
E 58.878 1.710181 5 55.87 60.17
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0090 ** -7.852914 -1.3150859
A - C -8.594 0.0001 *** -12.572895 -4.6151052
A - D -12.034 0.0000 *** -16.758213 -7.3097871
A - E -11.744 0.0000 *** -16.155720 -7.3322802
B - C -4.010 0.0193 * -7.278914 -0.7410859
B - D -7.450 0.0010 *** -11.861720 -3.0382802
B - E -7.160 0.0007 *** -11.138895 -3.1811052
C - D -3.440 0.0961 . -7.418895 0.5388948
C - E -3.150 0.0579 . -6.418914 0.1189141
D - E 0.290 0.8532 -2.978914 3.5589141
Prueba de Least significant difference
agricolae::LSD.test(modelo.dbca, trt ="trt", alpha =0.05, group =TRUE, main =NULL, console =TRUE)
Study: modelo.dbca ~ "trt"
LSD t Test for rdt
Mean Square Error: 5.944484
trt, means and individual ( 95 %) CI
rdt std r LCL UCL Min Max
A 47.134 2.840129 5 44.82253 49.44547 42.26 49.43
B 51.718 5.324328 5 49.40653 54.02947 44.41 57.31
C 55.728 2.229747 5 53.41653 58.03947 53.72 59.45
D 59.168 2.065798 5 56.85653 61.47947 55.87 60.89
E 58.878 1.710181 5 56.56653 61.18947 55.87 60.17
Alpha: 0.05 ; DF Error: 16
Critical Value of t: 2.119905
least Significant Difference: 3.268914
Treatments with the same letter are not significantly different.
rdt groups
D 59.168 a
E 58.878 ab
C 55.728 b
B 51.718 c
A 47.134 d
agricolae::LSD.test(modelo.dbca, trt ="trt", alpha =0.05, group =FALSE, main =NULL, console =TRUE)
Study: modelo.dbca ~ "trt"
LSD t Test for rdt
Mean Square Error: 5.944484
trt, means and individual ( 95 %) CI
rdt std r LCL UCL Min Max
A 47.134 2.840129 5 44.82253 49.44547 42.26 49.43
B 51.718 5.324328 5 49.40653 54.02947 44.41 57.31
C 55.728 2.229747 5 53.41653 58.03947 53.72 59.45
D 59.168 2.065798 5 56.85653 61.47947 55.87 60.89
E 58.878 1.710181 5 56.56653 61.18947 55.87 60.17
Alpha: 0.05 ; DF Error: 16
Critical Value of t: 2.119905
Comparison between treatments means
difference pvalue signif. LCL UCL
A - B -4.584 0.0090 ** -7.852914 -1.3150858
A - C -8.594 0.0000 *** -11.862914 -5.3250858
A - D -12.034 0.0000 *** -15.302914 -8.7650858
A - E -11.744 0.0000 *** -15.012914 -8.4750858
B - C -4.010 0.0193 * -7.278914 -0.7410858
B - D -7.450 0.0002 *** -10.718914 -4.1810858
B - E -7.160 0.0003 *** -10.428914 -3.8910858
C - D -3.440 0.0404 * -6.708914 -0.1710858
C - E -3.150 0.0579 . -6.418914 0.1189142
D - E 0.290 0.8532 -2.978914 3.5589142
